5.4 Testing your understanding of the two-server model Consider the city depicted in Figure P5.4. In this city emergency repair service is provided by two response units, prepositioned at (1, 0) and (-1, 0), respectively. Travel distance is right angle, with distance d between two points (x1, y1) and (x2, y2) equal to Repairs occur according to a spatial and temporal Poisson process with Unit 1's primary response area consists of all points east of the north-south boundary line that partitions the city. Unit 2's primary response area consists of all points west of that line. Note that the boundary, as drawn, is an equal travel-time boundary line. Whenever an emergency repair is reported from response area i (i = 1, 2), unit i is assigned, if available; otherwise, the other unit is assigned, if available. If neither unit is available, the call is lost (i.e., no queueing is allowed). Repair units travel at 100 miles/hr to and from the scene. On-scene service time is negative exponential with mean -1 = 10 hours. Upon completion of service, a unit always returns to its home location. Finally, o = 2 x 10-5 (emergencies per hour per square mile). Assume the system is operating in steady state. Average system-wide travel time to an emergency is . Using suitable approximations, or exact analysis if you prefer, find approximate nonzero values for 1 and 2, where m workload of unit m. The random variable Dx is defined to be the system-wide east-west (or west-east) travel distance for a random emergency. Sketch the pdf for Dx, again making suitable approximations. Approximately, what is the fraction of dispatches that are interresponse area dispatches? For parts (d)-(g) only, suppose that o = 10-2. This yields new values for 1, 2, , and so on. In considering each of these questions [parts (d)-(g)], assume that the total service-time distribution remains the same as that assumed in parts (a)-(c). Suppose the equal-travel-time boundary line is shifted a distance ( small) toward unit 1. Which is the appropriate response? As a result of the shift, will increase but |1 - 2| will decrease. As a result of the shift, will increase and |1 - 2| will increase. As a result of the shift, will decrease and |1 - 2| will decrease. As a result of the shift, will decrease but |1 - 2| will increase. As a result of the shift, we cannot tell which of the four possibilities above will apply. Now suppose that the equal-travel-time boundary line is shifted a distance ( small) toward unit 2. Dy is defined to be the system-wide north-south (or south-north) travel distance for a random emergency. As a result of the shift, E[Dy] will stay the same. E[Dy] will increase. E[Dy] will decrease. The behavior of E[Dy] cannot be determined. As in part (e), suppose that the equal-travel-time boundary line is shifted a distance ( small) toward unit 2. As a result of the shift: (1 + 2) will stay the same. (1 + 2) will increase. (1 + 2) will decrease. The behavior of (1 + 2) cannot be determined. Suppose that a third unit is added as a backup unit and is located at (x = 0, y = 0). This unit is assigned to any emergency that occurs when both units 1 and 2 are busy and unit 3 is available. Emergencies that arrive when all three units are busy are lost. Determine the workload of unit 3, 3.